The strong skew Hadamard conjecture: existence of skew Hadamard matrices of every order divisible by 4

A skew Hadamard matrix of order $n$ is a $\{\pm1\}$-matrix $H$ satisfying $HH^{\mathsf T}=nI$ and $H+H^{\mathsf T}=2I$, equivalently $H=I+C$ with $C$ a skew-symmetric conference matrix; such matrices can exist only in orders $1$, $2$, and multiples of $4$. The strong skew Hadamard conjecture, a strengthening of the classical Hadamard conjecture that traces back to Wallis (1971), asserts that a skew Hadamard matrix exists for every order divisible by $4$. The conjecture is open: constructions are known for every order below $356$, but order $356=4\cdot89$ and forty further orders up to $1200$ (404, 428, 476, 596, 612, 668, ...) have resisted every published method, while no obstruction or counterexample has ever been proposed. The conjecture is the engine behind optimal line packings (Fallon and Iverson, On the optimal arrangement of $2d$ lines in $\mathbb{C}^d$): a skew Hadamard matrix of order $d+1$ (or $2d$) yields a $d\times 2d$ complex equiangular tight frame, so the conjecture would settle attainability of the Welch bound for $2d$ lines in $\mathbb{C}^d$ in every dimension $d\not\equiv1\pmod4$.

Background

A Hadamard matrix of order $n$ is a square matrix $H$ with entries in $\{\pm1\}$ whose rows are mutually orthogonal, i.e. $HH^{\mathsf T}=nI$. A classical determinant argument shows such a matrix can exist only when $n=1$, $n=2$, or $n$ is a multiple of $4$, and the long-standing Hadamard conjecture asserts that matrices of all such orders exist; it remains open, although examples are known for every order below $668$. A skew Hadamard matrix is a Hadamard matrix of the special form $H=I+C$ with $C^{\mathsf T}=-C$, so that $H$ satisfies both $HH^{\mathsf T}=nI$ and $H+H^{\mathsf T}=2I$; the matrix $C=H-I$ is then a skew-symmetric conference matrix, a $\{\pm1\}$-matrix with zero diagonal and $CC^{\mathsf T}=(n-1)I$. Skew Hadamard matrices likewise exist only in orders $1$, $2$, and multiples of $4$. Deleting the first row and column of a normalized skew Hadamard matrix of order $n$ yields the adjacency matrix of a doubly regular tournament on $n-1$ vertices — a directed complete graph in which every vertex has out-degree $(n-2)/2$ and every two vertices have $(n-4)/4$ common out-neighbors — so the question below simultaneously asks for highly regular tournaments at every order $\equiv3\pmod4$.

Two classical sources of examples are Paley's 1933 construction, which produces a skew Hadamard matrix of order $q+1$ for every prime power $q\equiv3\pmod4$, and doubling constructions going back to Wallis (1971), which convert a skew Hadamard matrix of order $n$ into one of order $2n$; array-based constructions, supplementary difference sets, and orthogonal designs supply many further orders. Systematic tabulations, most recently the verified SageMath database of Cati and Pasechnik, show that these methods jointly construct skew Hadamard matrices of every order below $356$, while the orders $356=4\cdot89$, $404$, $428$, $476$, $596$, $612$, $668$, $708$, $712$, $716$, and thirty-one further orders up to $1200$ remain undecided.

The conjecture also has an analytic motivation in frame theory. For $n$ unit vectors in $\mathbb{C}^d$, the coherence (the largest absolute inner product between distinct vectors) is bounded below by the Welch bound $\sqrt{(n-d)/(d(n-1))}$, and equality forces the vectors to be equiangular and to form a tight frame — an equiangular tight frame (ETF) of size $d\times n$, which is the optimal line packing and the optimal unit-norm codebook for these parameters. For $2d$ lines in $\mathbb{C}^d$ (redundancy $2$) the bound reads $1/\sqrt{2d-1}$. Classical constructions give a $d\times 2d$ complex ETF whenever a symmetric or skew-symmetric conference matrix of order $2d$ exists, and Fallon and Iverson proved that a skew Hadamard matrix of order $d+1$ also suffices, via a variant of Hadamard doubling that converts any ETF of size $(d-1)/2\times d$ into one of size $d\times 2d$. Consequently, the strong skew Hadamard conjecture would prove that the Welch bound is attainable for $2d$ lines in $\mathbb{C}^d$ for every $d\not\equiv1\pmod4$, leaving only the residue class $d\equiv1\pmod4$ of the conjecture that a $d\times 2d$ complex ETF exists for every $d$.

Problem Statement

The strong skew Hadamard conjecture: for every positive integer $k$, there exists a skew Hadamard matrix of order $4k$, i.e. a matrix $H\in\{\pm1\}^{4k\times4k}$ satisfying

$HH^{\mathsf T}=4k\,I,\qquad H+H^{\mathsf T}=2I.$

Equivalently, for every $k$ there exists a skew-symmetric conference matrix of order $4k$ (namely $H-I$), or equivalently a doubly regular tournament on $4k-1$ vertices. The problem is to resolve this conjecture: either prove that skew Hadamard matrices exist at every order divisible by $4$, or prove that some order divisible by $4$ admits none. Constructions at individual orders — for instance at the currently undecided orders $356$, $404$, $428$, $476$, $596$, $612$, $668$, $708$, $712$, $716$, and thirty-one further orders up to $1200$ — advance the program but do not by themselves settle the conjecture, since an existence proof must cover all orders; a nonexistence proof at even a single order divisible by $4$, by contrast, would refute it.

References

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